A Few Class Diagram Kirchhoff Indicator

Similar to the Wiener index, the Kirchhoff index of a graph G is defined as the sum ofresistance distance between all pairs of vertices in G. This is a very important topological indexin Quantum Chemistry, which has been extensively studied.In1996, Gutman、Mohar[13]and Zhu、Klein、Lukovits[14]found independently the formulaof Kirchhoff index of a graph G in terms of the Laplacian spectrum of G, that is Lemma2.1.1inthis paper. This formula is suitable for the graphs whose Laplacian spectrum can be obtained.But we can not find directly the Laplacian spectrum of the graphs studied in this thesis. So, inthis thesis, we gives a special method to compute the Kirchhoff index of these graphs. It containsthe following two parts:First, the author in [10] found a special method to label vertices of some graphs. So, we usethis labeling method to deal with graphs studied in this thesis and obtain the Laplaciancharacteristic polynomials. In this part, we derived a formula of the Kirchhoff index of a graph Gin terms of the first derivative and the second derivative of the Laplacian characteristicpolynomials at zero, that is Lemma2.2.2. Using this Lemma, we get the Kirchhoff index ofgraphs, such as MS graphs、MC graphs、MCS graphs、MS graphs and MCS graphs which arebased on cycles or paths.Second, by the similar labeling method, we get the relationship between the Laplaciancharacteristic polynomial of the complement graph of related graph Kn Gand the Laplaciancharacteristic polynomial of graph G. We also derived a formula of the Kirchhoff index of agraph G in terms of the Laplacian characteristic polynomial of it’s complement graph G, thatis Lemma3.1.1. Using this Lemma, we get the formula of Kirchhoff index of the complement ofcycleC nor pathPn and some other graphs such as the MSR graphs、MCR graphs、MCSRgraphs、MSCR graphs、MCCR graphs、MSPR graphs and MCPR graphs.